The Mathematics of a Trip to Mars?

hakonhaugnes wonders: "Since trips to Mars seems commonplace (NASA has sent one every 26 months), I thought it made sense to try to understand how the interplanetary trajectory is calculated. NASA's page is deploringly void of intricate details. I found this excellent page, but it still left me feeling that I was missing something. Surely the calculus must go beyond two bodies (mars/earth)? (It seems there are commercial MATLAB scripts available but at $150 it went beyond the defensible to satisfy my curiosity). Are there any curious Slashdot readers with the usual great insight into how to calculate a trip to Mars?"

Comments

Much more than a 2-body problem ...

[IntelliTubbie]

Several of the people I work with in Caltech's Control and Dynamical Systems department work on celestial mechanics and calculating space flight trajectories -- and I can assure you, it's some pretty complicated stuff, involving invariant manifolds and (IIRC) patching together different three-body systems. There's a good popular article about this in Science News, and you can find more info (in as much detail as you'd like!) on Shane Ross' homepage.

Cheers,
IT

JPL has a good intro

[cunniff]

I was an intern at JPL a couple of decades ago, and they always started with a "porkchop plot" (or "butterfly plot") of possible trajectories and their energy requirements. Here is a webpage that documents that to some extent:

http://marsprogram.jpl.nasa.gov/spotlight/porkchop All.html

Re:JPL has a good intro

[Keebler71]

I did some graduate research/internship at JPL. They still use the porkchop plots which are published in volumes spanning the next decade or two. The "bible" at JPL (as far as I could tell) was The Interplanetary Mission Design Handbook Vol 1, Part 2. This is the document that I carry around with me to work or on travel if I think I am going to do a little research on the side (unfortunately my paying job has nothing to do with astrodynamics.) It covers pretty much all of the relevant equations for the various phases of an interplanetary mission (launch, transfer, arrival) as well as some other stuff. This is 35 pages of raw meat - little explanation, no derivations. Just the facts. I think the actual pork chop plots are in either other volumes or other parts of this volume (my paper copy had them right after this section).

Anyway, without at least some education in orbital mechanics/astrodynamics, the above ref will probably be a little overwhelming. To get up to speed I recommend the following:

• Fundamentals of Astrodynamics by Bate, Mueller & While. Undergrad text, should be no problem if you have had calculus.
• Fundamentals of Astrodynamics and Applications by Vallado. (This is usually referred to as "Vallado" - in fact I never even knew its title until I just looked it up!) This one is much more in-depth and is certainly found on the desk of anyone who does research in this field. Most of the stuff from the JPL handbook is in here, plus lots and lots of other stuff

Orbiter

[Anonymous Coward]

Orbiter is a great way to learn how those trips are done. It is a free simulator for windows and is available at www.orbitersim.com.
It has tools for calculating all sorts of interplanetary transfers and you can actually perform the flight from launch to landing on mars with all kinds of spacecraft.

Trajectory Math

[waynegoode]

I recently wrote some trajectory software for NASA. What I worked on is an approximation used for mission planning, not actual trajectories. I work with people who live and breathe this stuff and have worked on high-thrust and low-thrust trajectories for missions to the outer planets. I am mostly a software engineer, but I learned a lot from them while working on this project.

The key here is the energy required. Space travel is still dominated by propulsion. That is, the engines and the fuel they need, and the fuel needed to launch that fuel to orbit, etc., is where most of the cost is.

It is important to travel on a trajectory, called the transfer orbit, that requires the least energy. For a high thrust spacecraft, the minimum energy trajectory is called a Holman transfer. Simply, it is an orbit that just touches the orbits of both planets. The periapsis, the closest point to the sun, touches the orbit of the one planet and the apoapsis, the furtherest point, touches the other planet. For this to work, the destination planet needs to be half an orbit away when the spacecraft arrives. This is a lot easier to see in a picture.

For Earth to Mars, the spacecraft launches and then the thrusters fire to change the spacecraft's orbit of the sun from Earth's orbit to the transfer orbit. It then travels half of the transfer orbit and fires its thrusters to change its orbit to match Mars. This can be done by aerocapture, aerobraking or propulsion. The opportunity for a Holman transfer to Mars occurs every 26 years. It is based on the length of the orbit for the bodies being transferred between. The return trip also needs to be a Holman transfer to save fuel. The opportunity does not occur until many months after arrival. I forget the actual number. That is why Mars trips will have a long stay on Mars before returning.

Low thrust is different. Low thrust spacecraft thrust all or most of the time during the trip and the trajectory is more complicated. It is not usable for manned flight because it is to slow but is useful for unmanned spacecraft sometimes.

This is called Celestial Mechanics. When you add propulsion, it becomes Orbital Mechanics.

The best site I have found is NASA's Spacefligh Basics.

Also good is this site.

For explanation of gravity assists see this site.

Also see, Science World at Wolrram

Re:Trajectory Math

[hcg50a]

I think you mean 26 months, rather than years.

Mathematics is for Mathematicians

[Anonymous Coward]

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Re:Simple Newtonian

[sploxx]

Anyway he told us that most of the mathematical calculations that the Space Flight Center here in Houston use are the "simple" Newtonian laws of motion.
Sure. To use Einstein's general relativity would be overkill as the changes are too small.

But Newtons laws can get arbitrarily complex with the number of bodies that go into the equation.

One is newton's axiom.
Two is still easy and taught in school. Kepler ellipses etc. Together with the rocket equation (also only newton), it gives everything needed to go to earth orbit.

But.. three is not analytically solvable. From there, numerics takes over and this is still a very active field of research, still far from perfect. But they're surely good enough :-)

Fear and Fear Itself

[Doc Ruby]

The mathematical models for ballistic missiles isn't what's stopping "terrorists" from making them. What stops terrorists is that it's so much cheaper, faster, more reliable and easier to load a truck full of fertilizer and fuel oil, then blow up a skyscraper or maybe a bridge. Or just release a $25 video "around election time", which is about 18 months every 2 years (75% of the time). Both of which create terror, which is the entire point of terrorism.

There was a time when such math was secret, and strategic. But we caught up to the Soviets shortly after they tested that ballistic missile math on Sputnik, in the late 1950s. A half century later, our open society has proven more than a match for such "proprietary" losers. If we can stay that way, despite the exaggerated bugbears that people throw around to justify the secrecy that kills both science and liberty.

the Simple Answer and the Complex Answer

[everphilski]

The saying is "From low earth orbit, you are halfway to anywhere in the solar system." The delta-V (change in velocity) required to get to low earth orbit is about 7.6 m/s neglecting gravity and drag losses. The velocity to escape is about 13 m/s. Add in a little bit of velocity to correct your orbit to make it to Mars and it's about right, 14 m/s. (actually it'll be a bit more if you're launching from Kennedy, you have to get rid of that pesky inclination and that's an expensive maneuver, even combining it with the trans-martian injection it's expensive.

Here's the actual procedure.
1. surface to low earth orbit.
2. circularize low earth orbit. [hohmann transfer]
3. correct orbital parameters (longitude of ascending node, argument of periapsis, orbital inclination)
4. low earth orbit to trans-martian-injection [hohmann transfer]
(3 and 4 can be combined, to a point, in order to save delta-V.)
5. burn to circularize martian orbit [hohmann transfer]
6. correct orbital parameters (Same as 3)
7. Burn to descend to surface
The actual math is too much for a slashdot post. Sorry. If you are truly curious check out "Elements of Spacecraft Design" by Charles D. Brown.

-everphilski-